Fundamental solutions of the Knizhnik-Zamolodchikov equation of one variable and the Riemann-Hilbert problem

TL;DR

This paper connects the solutions of the KZ equation with multiple polylogarithms and zeta values through a Riemann-Hilbert problem, revealing new insights into the structure of fundamental solutions and their relation to the Drinfel'd associator.

Contribution

It introduces a novel approach linking the KZ equation's fundamental solutions to the Drinfel'd associator via a Riemann-Hilbert problem, unifying additive and multiplicative perspectives.

Findings

Derived multiple polylogarithms from multiple zeta values.

Established the Riemann-Hilbert problem as an inverse for the connection problem.

Linked the duality of the Drinfel'd associator to the solvability of the inverse problem.

Abstract

In this article, we derive multiple polylogarithms from multiple zeta values by using a recursive Riemann-Hilbert problem of additive type. Furthermore we show that this Riemann-Hilbert problem is regarded as an inverse problem for the connection problem of the KZ equation of one variable, so that the fundamental solutions to the equation are derived from the Drinfel'd associator by using a Riemann-Hilbert problem of multiplicative type. These results say that the duality relation for the Drinfel'd associator can be interpreted as the solvability condition for this inverse problem.